Modular invariants for real quadratic fields and Kloosterman sums

Abstract

We investigate the asymptotic distribution of integrals of the $j$-function that are associated to ideal classes in a real quadratic field. To estimate the error term in our asymptotic formula, we prove a bound for sums of Kloosterman sums of half-integral weight that is uniform in every parameter. To establish this estimate we prove a variant of Kuznetsov's formula where the spectral data is restricted to half-integral weight forms in the Kohnen plus space, and we apply Young's hybrid subconvexity estimates for twisted modular $L$-functions.

35 pages, v2 corrects a few errors from v1 (see the footnotes in Sections 1 and 2)

References in corpus (3)

• Weyl-type hybrid subconvexity bounds for twisted $L$-functions and Heegner points on shrinking sets
• A generalized cubic moment and the Petersson formula for newforms
• A Functional Equation for the Spectral Fourth Moment of Modular Hecke L-Functions

Cited by in corpus (7)

• Uniform bounds for Kloosterman sums of half-integral weight with applications
• Eisenstein series of even weight $k\ge 2$ and integral binary quadratic forms
• Formulas for non-holomorphic Eisenstein series and for the Riemann zeta function at odd integers
• Uniform bounds for Kloosterman sums of half-integral weight, same-sign case
• Effective estimates for traces of singular moduli
• A modular framework for generalized Hurwitz class numbers III
• Asymptotic distribution of traces of singular moduli